AISM 52, 351-366

## Robustness comparisons of some classes of location parameter estimators

### John R. Collins

Department of Mathematics and Statistics, University of
Calgary, Calgary, Alberta, Canada T2N 1N4

(Received May 13, 1997; revised August 28, 1998)

Abstract.
Asymptotic biases and variances of *M*-, *L*- and
*R*-estimators of a location parameter are compared under
*epsilon*-contamination of the known error distribution *F*_{0} by an unknown
(and possibly asymmetric) distribution. For each *epsilon*-contamination
neighborhood of *F*_{0}, the corresponding
*M*-, *L*- and *R*-estimators which are asymptotically efficient at the least
informative distribution are compared under asymmetric
*epsilon*-contamination. Three scale-invariant versions of the
*M*-estimator are studied: (i) one using the interquartile range as a
preliminary estimator of scale: (ii) another using the median absolute
deviation as a preliminary estimator of scale; and (iii) simultaneous
*M*-estimation of location and scale by Huber's Proposal 2. A question
considered for each case is: when are the maximal asymptotic biases and
variances under asymmetric *epsilon*-contamination attained by unit point
mass contamination at \infty? Numerical results for the case of the
*epsilon*-contaminated normal distribution show that the
*L*-estimators have generally better performance (for small to moderate
values of *epsilon*) than all three of the scale-invariant *M*-estimators
studied.

Key words and phrases:
Robust estimation, *M*-, *L*- and
*R*-estimators, asymptotic biases, asymptotic variances, asymmetric
contamination.

**Source**
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